
theorem GRCY212:
  for k being Element of NAT, G being finite Group, a being Element of G
  holds gr{a} = gr{(a |^ k)} iff k gcd (ord a) = 1
  proof
    let k be Element of NAT, G be finite Group,
    a be Element of G;
    set n = ord a;
    reconsider a0 = a as Element of gr{a} by GR_CY_2:2, STRUCT_0:def 5;
    A11: gr{a0} = gr{a} by GR_CY_2:3;
    card (gr{a}) = n by GR_CY_1:7;
    then gr{a} = gr{(a0 |^ k)} iff k gcd n = 1 by A11, GR_CY_2:12;
    hence thesis by GROUP_4:2, GR_CY_2:3;
  end;
